Solve MATLAB Matrix Problem: Find Basis of Kernel in MATLAB

[Ermol]

1. It's a MATLAB project. given a matrix of the type
A (mod p)
where p prime (and for now say the matrix is square, though i don't think this is that relevant) i need to write a program in Matlab that will calculate the basis of the kernel of the matrix. Obviously the null program wouldn't work.

Homework Equations

Prior to this I wrote the program that finds the row echelon form and the rank of any matrix of this form.

The Attempt at a Solution

What I'm trying to do is if i have a matrix of the form:

[1 . . . .
0 1 . . .
0 0 0 1 .
0 0 0 0 0
0 0 0 0 0]

or something along those lines.

Take any such matrix A, and we want to find the basis for the space of all x such that Ax=0. So we can express:
x₄ in terms of x₅
x₂ in terms of x₃ and x₅
x₁ in terms of x₃ and x₅
That was basically the hint we got, so need to express the x_i corresponding to the first elements in each row that has non-zero elements in terms of the other x_j.

I got this so far, though it's obviously incomplete


[m,n] = size(A);
A=echelon(A,p);

i=1:r;
j=r+1:m;
B=A(i,:)

br=B(r,:)
x(r) = br(j)*x(j) (*)


My first question is how to define an x s.t. this works? I would normally start with x=zeros(r,1), but it won't work with the code above. I would then define x(r) as in (*). Then x(r-1) and so on up til x=1. But i don't know what exactly to do.

 

[kurt.physics]

I'm assuming that I would need loops to do this, but I'm not sure how to go about it. Any help would be greatly appreciated, thanks!